Spectral theory is concerned with properties of linear operators in Banach spaces.
Given a linear operator in a complex Banach space with domain the spectrum is the set of all complex such that is not an isomorphism (here is equipped with the graph norm). Most prominent and already known from finite dimensions (linear algebra) are eigenvalues , but in infinite dimensions new phenomena arise.
The complement of in is called the resolvent set of .
Central topics in spectral theory are properties of the spectrum including investigation of eigenvalues and eigenvectors, properties of the resolvent map , decomposition of the space in invariant subspaces and existence of functional calculi for .
In the lecture we shall study in particular
- spectrum and resolvent for bounded and unbounded operators,
- spectral properties of compact operators in Banach spaces,
- the spectral theorem for self adjoint operators in Hilbert space,
- applications to differential operators and boundary value problems.
|Summary of the lectures|
J.B. Conway: A Course in Functional Analysis, Springer.
E.B. Davies: Spectral Theory and Differential Operators, Cambridge University Press.
N. Dunford, J.T. Schwartz: Linear Operators, Part I: General Theory, Wiley.
D.E. Edmunds, W.D. Evans: Spectral Theory and Differential Operators, Oxford University Press.
T. Kato: Perturbation Theory of Linear Operators, Springer.
S. Lang: Real and Functional Analysis, Springer.
R. Meise, D. Vogt: Introduction to Functional Analysis, Oxford, Clarendon Press.
W. Rudin: Functional Analysis, McGraw-Hill.
D. Werner: Funktionalanalysis, Springer.